Quantum Probability's Algebraic Origin

TL;DR

This paper explores the algebraic foundations of quantum probabilities, showing they differ fundamentally from classical probabilities and introducing a new, physically meaningful definition of transition probabilities that reveal quantum indeterminacy.

Contribution

It introduces a general algebraic definition of transition probabilities that extends beyond traditional quantum states, linking algebraic structures to probability values without relying on wave functions.

Findings

Quantum probabilities have an algebraic origin distinct from classical probabilities.

The new transition probability encompasses known and novel physically meaningful cases.

It demonstrates how algebraic structures determine specific quantum probability values.

Abstract

Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg's and others' uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.